Christoffel symbol transformation law

Question

It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is:

$$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta \gamma}{\partial x^\beta \over \partial \tilde x^\nu}{\partial x^\gamma \over \partial \tilde x^\kappa} + {\partial ^2 x^\alpha \over \partial \tilde x^\nu \partial \tilde x^\kappa} \right ]$$

Is there any way to prove this rule using only the definition of the Christoffel via the metric tensor? That is, using:

$$ \Gamma^\mu _{\nu\kappa} = \frac{1}{2}g^{\mu\lambda}\left(g_{\lambda\kappa,\nu}+g_{\nu\lambda,\kappa}-g_{\nu\kappa,\lambda} \right)$$

All proofs have I've seen of the transformation law involve another method.

Answer 1

This is very straightforward, just substitute the transformation rules and collect the terms.

Here are some details.

The inverse metric transforms, as we know, by the rule: $$ g^{\mu \lambda} = \frac{\partial{\bar{x}}^\mu}{\partial{x}^\alpha} \frac{\partial{\bar{x}}^\lambda}{\partial{x}^\delta} g^{\alpha \delta} $$

The partial derivatives need some calculations that can be presented as $$ \begin{align*} g_{\lambda \kappa , \nu} & = \frac{\partial}{\partial{\bar{x}^\nu}} \Big( \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} g_{\delta \gamma} \Big) \\ &= \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} \frac{\partial{x^\beta}}{\partial{\bar{x}^\nu}} g_{\delta \gamma , \beta} + g_{\delta \gamma} \frac{\partial}{\partial{\bar{x}^\nu}} \Big( \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} \Big) \end{align*} $$

Similarly, $$ g_{\nu \lambda , \kappa} = \frac{\partial{x^\beta}}{\partial{\bar{x}^\nu}} \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} g_{\beta \delta , \gamma} + g_{\beta \delta} \frac{\partial}{\partial{\bar{x}^\kappa}} \Big( \frac{\partial{x^\beta}}{\partial{\bar{x}^\nu}} \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} \Big) $$ and $$ g_{\nu \kappa , \lambda} = \frac{\partial{x^\beta}}{\partial{\bar{x}^\nu}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} \frac{\partial{x^\delta}}{\partial{\bar{x}^\lambda}} g_{\beta \gamma , \delta} + g_{\beta \gamma} \frac{\partial}{\partial{\bar{x}^\lambda}} \Big( \frac{\partial{x^\beta}}{\partial{\bar{x}^\nu}} \frac{\partial{x^\gamma}}{\partial{\bar{x}^\kappa}} \Big) $$

Substituting these identities into your "definition" $$ \Gamma^\mu _{\nu\kappa} = \frac{1}{2}g^{\mu\lambda}\left(g_{\lambda\kappa,\nu}+g_{\nu\lambda,\kappa}-g_{\nu\kappa,\lambda} \right) $$ and taking into account that $$ \Gamma^\alpha _{\beta \gamma} = \frac{1}{2}g^{\alpha \delta}\left(g_{\delta \gamma , \beta}+g_{\beta \delta , \gamma} - g_{\beta \gamma , \delta} \right) $$ it is not difficult now to show the required transformation rule for the Christoffel symbols.

Answer 2

Let $(U, x^1, \ldots, x^n)$ be a chart of Riemannian manifold $M$, smooth coordinate chang $(U, g_{\mu\nu}, x^1, \ldots, x^n) \to (U, g'_{\mu'\nu'}, y^1, \ldots, y^n)$. Then $\partial^x_\mu =\frac{\partial}{\partial x^\mu} = \frac{\partial y ^ {\mu'}}{\partial x^\mu} \frac{\partial}{\partial y^{\mu'}} = \frac{\partial y ^ {\mu'}}{\partial x^\mu} \partial^y_{\mu'}$, $\partial^x_\nu = \frac{\partial y ^ {\nu'}}{\partial x^\nu} \partial^y_{\nu'}$.

Inner product is invariant under coordinate change.

For $v, w\in T_p U, v = v^\mu \partial^x_\mu = v^\mu \frac{\partial y ^ {\mu'}}{\partial x^\mu} \partial^y_{\mu'}, w = w^\nu \partial^x_\nu = w^\nu \frac{\partial y ^ {\nu'}}{\partial x^\nu} \partial^y_{\nu'}$.

$\langle v, w\rangle = g_{\mu\nu}v^\mu w^\nu = g'_{\mu'\nu'}\frac{\partial y ^ {\mu'}}{\partial x^\mu}\frac{\partial y ^ {\nu'}}{\partial x^\nu} v^\mu w^\nu $, thus $g'_{\mu'\nu'} = \frac{\partial x^\mu}{\partial y ^ {\mu'}} \frac{\partial x^\nu}{\partial y ^ {\nu'}}g_{\mu\nu}$. $g$ is a tensor.

$\Gamma_{\mu \nu}^{\lambda}=\frac{1}{2} g^{\lambda \rho}(\partial_{\mu}g_{\nu \rho} +\partial_{\nu}g_{\rho \mu}-\partial_{\rho}g_{\mu \nu}), {\Gamma'}_{\mu' \nu'}^{\lambda'}=\frac{1}{2} {g'}^{\lambda' \rho'}(\partial_{\mu'} {g'}_{\nu' \rho'} +\partial_{\nu'} {g'}_{\rho' \mu'}-\partial_{\rho'}{g'}_{\mu' \nu'})$.

$\partial_{\alpha '}g_{\beta ' \gamma '}' =\frac{\partial x^{\alpha } }{\partial y^{\alpha '} }\frac{\partial x^{\beta } }{\partial y^{\beta '} }\frac{\partial x^{\gamma } }{\partial y^{\gamma'}} \partial_\alpha g_{\beta \gamma} + g_{\beta \gamma} (\frac{\partial x^{\gamma} }{\partial y^{\gamma'} }\frac{\partial^{2} x^{\beta } }{\partial y^{\alpha '}\partial y^{\beta '} }+ \frac{\partial x^{\gamma } }{\partial y^{\beta '} }\frac{\partial^{2} x^{\beta } }{\partial y^{\alpha '}\partial y^{\gamma '}})$ \begin{align}\notag &\partial_{\mu'} {g'}_{\nu' \rho'} + \partial_{\nu'} {g'}_{\rho' \mu'}-\partial_{\rho'}{g'}_{\mu' \nu'}\\ \notag&=\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '} }\frac{\partial x^{\rho } }{\partial y^{\rho'}} \partial_\mu g_{\nu \rho} + g_{\nu \rho} (\frac{\partial x^{\rho} }{\partial y^{\rho'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\nu '} }+ \frac{\partial x^{\rho } }{\partial y^{\nu '} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\rho '}})\\ \notag&+ \frac{\partial x^{\nu } }{\partial y^{\nu '} }\frac{\partial x^{\rho } }{\partial y^{\rho '} }\frac{\partial x^{\mu } }{\partial y^{\mu'}} \partial_\nu g_{\rho \mu} + g_{\rho \mu} (\frac{\partial x^{\mu} }{\partial y^{\mu'} }\frac{\partial^{2} x^{\rho } }{\partial y^{\nu '}\partial y^{\rho '} }+ \frac{\partial x^{\mu} }{\partial y^{\rho '} }\frac{\partial^{2} x^{\rho } }{\partial y^{\nu '}\partial y^{\mu '}})\\ \notag&-\frac{\partial x^{\rho } }{\partial y^{\rho '} }\frac{\partial x^{\mu } }{\partial y^{\mu '} }\frac{\partial x^{\nu } }{\partial y^{\nu'}} \partial_\rho g_{\mu \nu} - g_{\mu\nu} (\frac{\partial x^{\nu} }{\partial y^{\nu'} }\frac{\partial^{2} x^{\mu } }{\partial y^{\rho'}\partial y^{\mu'} }+ \frac{\partial x^{\nu} }{\partial y^{\mu'} }\frac{\partial^{2} x^{\mu} }{\partial y^{\rho '}\partial y^{\nu'}})\\ \notag&=\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '} }\frac{\partial x^{\rho } }{\partial y^{\rho'}}(\partial_\mu g_{\nu \rho}+\partial_\nu g_{\rho \mu}-\partial_\rho g_{\mu \nu})+g_{\nu \rho} (\frac{\partial x^{\rho} }{\partial y^{\rho'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\nu '} }+ \frac{\partial x^{\rho } }{\partial y^{\nu '} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\rho '}}\\ \notag&\quad+\frac{\partial x^{\rho} }{\partial y^{\mu'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\nu '}\partial y^{\rho '} }+ \frac{\partial x^{\rho} }{\partial y^{\rho '} }\frac{\partial^{2} x^{\nu } }{\partial y^{\nu '}\partial y^{\mu '}} -\frac{\partial x^{\rho} }{\partial y^{\nu'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\rho'}\partial y^{\mu'} }- \frac{\partial x^{\rho} }{\partial y^{\mu'} }\frac{\partial^{2} x^{\nu} }{\partial y^{\rho '}\partial y^{\nu'}})\\ \notag&=\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '} }\frac{\partial x^{\rho } }{\partial y^{\rho'}}(\partial_\mu g_{\nu \rho}+\partial_\nu g_{\rho \mu}-\partial_\rho g_{\mu \nu}) +2 g_{\nu \rho} \frac{\partial x^{\rho} }{\partial y^{\rho'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\nu '} } \end{align} \begin{align}\notag {\Gamma'}_{\mu' \nu'}^{\lambda'} &=\frac{1}{2} {g'}^{\lambda' \rho'}(\partial_{\mu'} {g'}_{\nu' \rho'} +\partial_{\nu'} {g'}_{\rho' \mu'}-\partial_{\rho'}{g'}_{\mu' \nu'})\\ \notag&=\frac{1}{2} (\frac{\partial y ^ {\lambda'}}{\partial x^\lambda} \frac{\partial y ^ {\rho'}}{\partial x^\beta}g^{\lambda\beta})(\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '} }\frac{\partial x^{\rho } }{\partial y^{\rho'}}(\partial_\mu g_{\nu \rho}+\partial_\nu g_{\rho \mu}-\partial_\rho g_{\mu \nu}) +2 g_{\nu \rho} \frac{\partial x^{\rho} }{\partial y^{\rho'} }\frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\nu '} })\\ \notag&=\frac{1}{2}g^{\lambda\beta}\delta_\beta^\rho \frac{\partial y ^ {\lambda'}}{\partial x^\lambda}\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '}} (\partial_\mu g_{\nu \rho}+\partial_\nu g_{\rho \mu}-\partial_\rho g_{\mu \nu})+ g_{\nu \rho}g^{\lambda\beta}\delta_\beta^\rho \frac{\partial y ^ {\lambda'}}{\partial x^\lambda} \frac{\partial^{2} x^{\nu } }{\partial y^{\mu '}\partial y^{\nu '}}\\ \notag&=\frac{\partial y ^ {\lambda'}}{\partial x^\lambda}\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '}} \frac{1}{2}g^{\lambda\rho}(\partial_\mu g_{\nu \rho}+\partial_\nu g_{\rho \mu}-\partial_\rho g_{\mu \nu})+\frac{\partial y ^ {\lambda'}}{\partial x^\lambda} \frac{\partial^{2} x^{\lambda } }{\partial y^{\mu '}\partial y^{\nu '}}\\ \notag&=\frac{\partial y ^ {\lambda'}}{\partial x^\lambda}\frac{\partial x^{\mu } }{\partial y^{\mu'} }\frac{\partial x^{\nu } }{\partial y^{\nu '}}\Gamma^\lambda_{\mu\nu}+\frac{\partial y ^ {\lambda'}}{\partial x^\lambda} \frac{\partial^{2} x^{\lambda } }{\partial y^{\mu '}\partial y^{\nu '}} \end{align} Thus Christoffel symbol is not a tensor.